ウェーブレット理論と工学への応用 - AIMaP...The original chromatic scale All the...

1 12 5 13:20 18:00
1 12 6 10:00 13:00
Wavelet theory and its applications to engineering
OKU & AIMaP 2019 Workshop
2019 12 5 – 12 6
1 (A)
2019 12 5 13:20 – 18:00
13:20–13:30
()

()

DWT

DES

10:00 – 11:00

Mellin
()

543-0054 4-88 (06)6775-6611
JR 10
JR 5
https://aimap.imi.kyushu-u.ac.jp/wp/

Contents 1. Preliminaries.
2. We introduce our proposed Hilbert transform pairs of orthonormal bases of chromatic-scale wavelets.
3. Comparing between Gabor and our wavelets, we appeal why we want to use the Gabor wavelet in the discrete wavelet transform.
4. For this aim, we propose “Tight wavelet frame using complex wavelet designed in free shape of frequency domain”.
5. We construct the tight wavelet frame using an approximate Gabor wavelet.
6. The dual wavelet frame using an approximate Gabor wavelet. The contents are like these.
1
2
OKU & AIMaP 2019 Workshop on Wavelet theory and its applications to engineering 1

fff ,
Our proposed “Hilbert transform pairs of orthonormal bases of chromatic-scale wavelets”
The original chromatic scale

R
j
j
3
4
OKU & AIMaP 2019 Workshop on Wavelet theory and its applications to engineering2

0 500 1 103 1
0
1
n
2
4
2
4
2
4
2
4
2
4
2
4
2
4
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4
2
4
50 0 50 0.4
0,1ˆ
1
2
3
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5
Imag.
Real
t0,1
t
5
6

1
2
3
4
5
0.2
0
0.2
0.4
1
2
3
4
Gabor wavelet
Trapezium=
Gaussian function
Imag.
Real
tG
Gˆ

0.2
0
0.2
0.4
1
2
3
←→
7
8

1
2
3
4
5
0.2
0
0.2
0.4
1
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Gabor wavelet
Trapezium=
Gaussian function
0.5
0
0.5
1
0.5
1
1.5
0.5
0
0.5
1
0.5
1
1.5

Complex wavelet based on Meyer wavelet Trapezium= Considering the uncertainty principle …
Time domain
Frequency domain

9
10

For this aim, we propose the following method:
1. We design the original mother wavelet having free shape on the frequency domain.
2. Using the original mother wavelet, we construct an approximate tight wavelet frame.
3. Based on it, we construct a tight wavelet frame with minor modification.
We want to use the Gabor wavelet for digital transform!
The tight wavelet frame using an approximate Gabor wavelet
tf tf Transform Inv. transform
tf N nj
P nj dd ,, ,
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A
Rem nj
N nj
j n
Rem nj
P njRem
tdtd A
tf ,,,, 1
N nj
11
12

7
Theorem 1 for tight wavelet frames The Fourier transform of has a compact support of length as follows:
,,,ˆsupp 1010
.20 p is a constant real number.
We define the following transform of . 0p
R2Ltf
RR 21 LLt
,ˆ0 sup R
fWF
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1

Origˆ
OrigˆThe original mother wavelet must be normalized and have a compact support on the frequency domain, but its shape is free!
We recommend the positive real function value, easy to treat
13
14

1
2
3
1
2
3
1
2
Time domain
Time domain
Time domain
Frequency domain
Frequency domain
Frequency domain
An example of the original mother wavelet based on Gabor wavelet tOrig
,
'
'
1
2
3
Gˆ
15
16

9
The distance p>0 between wavelets in level 0 and the dilation a>1
,20 p
.1 0
a
Example of p>0 and a>1 based on Gabor wavelet can be set based on .2,3, 10
,1p
4 2 0 2 4 0
2
4
6
8
., ,,
domain
17
18

., ,,
,, nptaat jOrigjOrig nj
.ˆ fOrigfWF Orig
2
4
6
8
Orig nj
Orig nj
j n
Orig nj
Orig njOrig
tftf A
2
4
6
8
Orig
4 2 0 2 4 0
2
4
6
8
2
4
6
8
2
4
6
8

19
20

Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A

constructs a tight wavelet frame as follows: Znjtt Rem nj
Rem nj ,:, ,,
The third step is the construction of a tight wavelet frame using a remaking mother wavelet
0 100 200 300 400 0
2
4
2
4
2
4
0 500 1 103 1
0
1
n
2
4
2
4
n
21
22

1
2
3
1
2
3
1
2
3
Gˆ
1
2
0
1
2
3
1
2
3
Gˆ
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24

0.1
0.2
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0.5
0.2
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0.8
1
0.2
0.4
0.6
0.8
1
1
2
3
1
2
3
1
2
3

The dual orthogonal wavelet “B spline wavelet of order 4”
0 5 10 0.3
Orthogonal relation nnjjnjnj ,,,, ~,
25
26

0.2
0.4
0.6
0.8
1
2
4
6
8
10
0.1
0.2
0.3
The dual orthogonal wavelet “B spline wavelet of order 4”
0 5 10 0.3
tf N nj
P nj dd ,, ,
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Rem nj
Rem nj
j n
Rem nj
Rem njRem
tftf A
Rem nj
N nj
j n
Rem nj
P njRem
tdtd A
tf ,,,, 1
N nj
27
28

Orig nj
M nj
Orig nj
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
Orig nj
Orig nj
j n
Orig nj
Orig njOrig
tftf A
Orig nj
N nj
j n
Orig nj
P njOrig
tdtd A
tf ,,,, 1
N nj
,,ˆ~ 10
Orig Orig
Orig Orig
A Remaking
Theorem 1 for tight wavelet frames The Fourier transform of has a compact support of length as follows:
,,,ˆsupp 1010
.20 p is a constant real number.
R2Ltf
RR 21 LLt
,ˆ0 sup R
fWF
29
30

16
Theorem 2 for dual wavelet frames The Fourier transform of and have a compact support of length as follows:
RR 21~ LLt
,~0 sup R
,:~~ Z nnpttn
R2Ltf
.ˆˆ~1,~ f
0.1
0.2
0.3
0.4
0.5
1
2
3
1
2
3
tOrig~ tOrig
t t
OrigOrig
31
32

0.2
0.4
0.6
0.8
1
1
2
3
1
2
3
tOrig~ tOrig
t t
0.2
0.4
0.6
0.8
1
1
2
3
1
2
3
tOrig~ tOrig
t t
OrigOrig
33
34

0
2
4
2
0
2
4
2
0
2
4
2
0 500 1 103 1
0
1
n
2
0
2
4
2
0
2
4
2
0 500 1 103 1
0
1
n
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36

2
4
0
2
4
6
2
4
2
4
0 100 200 300 0
2
4
0 500 1 103 1
0
1
n
0.5
1
1.5
2
5 0 5 1
37
38

0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Mˆ
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Mˆ
OrigOrig
39
40

1
2
3
4
0.5
1
1.5
2
Orig~
0.5
1
1.5
2
1
2
3
4
0.5
1
1.5
2
41
42

2
4
The analysis by the dual wavelet frame based on Meyer
0 500 1 103 1
0
1
n
1
2
3
n
Conclusion 1. We introduced our proposed Hilbert transform pairs of
orthonormal bases of chromatic-scale wavelets.
2. We introduced the Gabor wavelets.
3. We introduced the tight wavelet frame using complex wavelet designed in free shape on frequency domain.
4. We constructed the tight wavelet frame using an approximate Gabor wavelet.
5. We constructed the dual wavelet frame using an approximate Gabor wavelet.
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2019/12/5 AIMap2019(N. Ikawa) 2
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AIMap2019(N. Ikawa) 32019/12/5
AIMap2019(N. Ikawa) 4
4
20dB
2019/12/5
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4

• (Auditory Brainstem Response : ABR)
• auditory steadystate response : ASSR
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40-Hz ASSR 80-Hz ASSR
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40Hz ASSR Kawase 2015, Audiology Japan 581 p.47
AIMap2019(N. Ikawa) 92019/12/5
– MB11
– MASTER
–
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An example of MASTER output window:
- 70 dB : 512 epochs --> 32 sweeps - 60 dB : 474 epochs --> 29 sweeps - 80 dB : 184 epochs --> 11 sweeps (1 sweep = 16 epochs)
(Aoyagi[1])
80Hz-ASSR
13
In the case of 80-Hz ASSR, MASTER system uses both relationship between the carrier frequency of input stimuli and the modulation frequency of cochlea responses. 2019/12/5
Diatec HP https://www.diatec- diagnostics.jp/solutions/products/abr/interacoustics-eclipse-complete-solution#
•CEChirp® •ASSR8
=>
=>
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Diatec HP https://www.diatec- diagnostics.jp/solutions/products/abr/interacoustics-eclipse-complete-solution#
2019/12/5
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CF= 125 8000 Hz, MF0100H
Sound intensity: 0 90 dB (nHL)
CF= 1000 Hz, MF40H Intensity: 80 dB (nHL)
2019/12/5
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Averaged EEG data:
•β1430Hz
βγ28Hz

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D1
D2
D3
D4
D5
D6
D7
A7
S

D4
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Here, we put , 2,
Using MATLAB function cmorwavf(Lb, Ub, U, N, fb, fc), where N = 1000; Lb = 5; Ub = 5; fb = 1.5; fc = 1;
2019/12/5
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40-Hz ASSR(70dB, 50dB, 30dB) 40-Hz ASSR
2019/12/5 AIMap2019(N. Ikawa) 33
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[1] , , , 2004, . [2] N. Ikawa, T. Yahagi and H. Jiang, "Waveform analysis based on latency frequency characteristics of auditory brainstem response using wavelet transform," Journal of Signal Processing, Vol. 9, No.6, pp. 505518, 2005.11. [3] M. S. John, et al., MASTER: aWindow program for recording multiple auditory steadystate responses, Comput. Methods Programs Biomed., 61, 125–150, 1998. [4] , PXI4461 , CFME, 2009. [5] , , , “40Hz ”, JSIAM, 47–48, 2011 [6] , , , “
”,, Vol.27 No.2, pp.216238, 2017.6.25.
[7] , , ,“ ” ,
http://www.osakakyoiku.ac.jp/~morimoto/WSPRO/wavelet2013proceedings.pdf, 25 ,2013.11.23.
[8], , ,“ ”, JSIAM, 2501_2018080520080097313.pdf, 2018.9..
AIMap2019(N. Ikawa) 372019/12/5
AIMap2019(N. Ikawa) 382019/12/5
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2019/12/6
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• 18 6
DWT DES

2019/12/6
2

•
• DWT

• • • •
2019/12/6
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DESAES
• DES DES AES
• AESAdvanced Encryption StandardAES- 128AES-192AES-2563
AES
• AES AES
2019/12/6
4
RSA
• RSA
RSA VS AES
• RSA
2019/12/6
5
/ • ”SSL”
• ””SSL
•

•
•
•
•
•
2019/12/6
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SSL
• Web
• SSL
• SSL
WordPress (PHP) • Web

•
• WordPress
https://techacademy.jp/briefing-lp-wordpress-s?utm_source=yahoo&utm_medium=cpc&utm_campaign =03_briefing_wordpress&yclid =YSS.1000002331.EAIaIQobChMIkeD5076d5gIVQ7aWCh1vbQuaEAAYASAAEgItfPD_BwE
2019/12/6
7
• • 1DES() • 1643

•
• DES
2019/12/6
8
• DESDES3 DES
• UNIX DES
DES
• DES 3 8 DES 3 DATA 8
• • 2 • 3 2 • DES
• 3 • 2 • 2
2019/12/6
9
• 3DWT RGB 4
• 4R GBHH
• 5IDWT
.docx.xlsx.pdf .txtK1K2 K3
2019/12/6
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• 2HH
• 43DES

2019/12/6
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• (a) Airplane.png • (b) Arctichare.png • (c) Baboon.png • (d) Boat.png • (e) Boy.bmp • (f) Cat.png • (g) Fruits.png • (h) Frymire.png • (i) Lena.png • (j) Peppers.png
TYPES OF SUPPORTED DOCUMENTS

• 12 KB docx 9.67 KBxlsx22.3 KBpdf161 txt
2019/12/6
12
PC: Let’s Note CF-R9
• CPU: Intel Core i7 (1.07GHz) • RAM: 4GB • 32 bit OS • X64 Base Processor

2019/12/6
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• MSE PSNR
Airplane 512x512
MSE 18 14 - 2
MSE 18 16 - 2
512x512 MSE 18 15 - 2 PSNR 23 25 - 43
Boy 768x512
MSE 14 11 22 1 PSNR 26 27 21 46
Cat 490x733
MSE 14 12 23 2 PSNR 25 27 21 45
Fruits 512x512
Frymire 1118x1106
PSNR 34 35 31 52 Lena
512x512 MSE 17 15 - 2 PSNR 23 25 - 43
Peppers 512x512
MSE 17 15 - 2
PSNR 23 25 - 43
2019/12/6
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Airplane.png HH 50 airplane.encripsi.nilai.50.png Original Document 43 dB
2019/12/6
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Airplane.txt.png Brightness 30% Brightness.30%.png Original Document
Airplane.txt.png Brightness -150% Brightness.minus150%.png Original Document
Airplane.txt.png Contrast 100% Contrast.100%.png Original Document
Airplane.txt.png Contrast -50% Contrast.minus50%.png Original Document
Airplane.txt.png Crop Up Crop.atas.png Original Document
Airplane.txt.png Crop Down Crop.bawah.png Blank Document Airplane.txt.png Crop Right Crop.kanan.png “Can’t be Extracted”
Airplane.txt.png Crop Left Crop.kiri.png "The Key That You Entered Is Incorrect"
Airplane.txt.png Resize 400x400 Resize.400x400.png “Can’t be Extracted”
Airplane.txt.png Resize 514x514 Resize.514x514.png “Can’t be Extracted” Airplane.txt.png Rotate 90o CW Rotate.kanan.png Blank Document Airplane.txt.png Rotate 90o CCW Rotate.kiri.png “Can’t be Extracted” Airplane.txt.png Rotate 180o Rotate.penuh.png “Can’t be Extracted”
2019/12/6
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Original Encryption Result
2019/12/6
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2019/12/6
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• HLHHMSEPSNR
• MSEPSNR Insertion Value50
•
2019/12/6
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References
• Kohei Arai, Kaname Seto, Data Hiding Based on Wavelet Multi-resolution Analysis, Journal of Visualization Society of Japan, Vol.22, Suppl.No.1, 229-232, 2002.
• Kohei Arai, Kaname Seto, Data Hiding Based on Multi- resolution Analysis Utilizing Information Content Concentrations by Means of Eigen Value decomposition, Journal of Visualization Society of Japan, Vol.23, No.8, pp.72- 79,2003.
• Kohei Arai, Kaname Seto, Information Hiding Method Based on Coordinate Conversion, Journal of Visualization Society of Japan, 25, Suppl.No.1, 55-58,(2005)
• Kohei Arai, Kaname Seto, Data hiding based on Multi-Resolution Analysis taking into account scanning of the embedded image for improvement of invisibility, Journal of Visualization Society of Japan, 29, Suppl.1, 167-170, 2009.
• Kohei Arai and Yuji Yamada, Improvement of secret image invisibility in circulation image with Dyadic wavelet based data hiding with run-length coding, International Journal of Advanced Computer Science and Applications, 2, 7, 33-40, 2011
• Kohei Arai, Method for data hiding based on Legal 5/2 (Cohen- Daubechies-Feauveau: CDF 5/3) wavelet with data compression and random scanning of secret imagery data, International Journal of Wavelets Multi Solution and Information Processing, 11, 4, 1-18, B60006 World Scientific Publishing Company, DOI: I01142/SO219691313600060, 1360006-1, 2013.
2019/12/6
20
• Kohei Arai, Data Hiding Method Replacing LSB of Hidden Portion for Secrete Image with Run-Length Coded Image, International Journal of Advanced Research on Artificial Intelligence, 5, 12, 8-16, 2016.
• Cahya Rahmed Kohei Arai, Arief Prasetyo, Noriza Arigki, Noble Method for Data Hiding using Steganography Discrete Wavelet Transformation and Cryptography Triple Data Encryption Standard: DES, International Journal of Advanced Computer Science and Applications: IJACSA, 9, 11, 261-266, 2018.
Mf(ω) =
f(t) tωdt (1)
ω ∈ R+ ω n Mellin n
Mf(ω) C
f(t) = 1
1.1 Mellin
Mellin (1) Roger Godement Springer “Analysis III” [1] Wolfram MathWorld [2] Wikipedia [3] Mellin
Mf(s) = φ(s) =
1
Mellin

2019 12 5
1 Mellin R+ := {t ∈ R | t ≥ 0} f(t) Mellin
c φ(s) φ(s) φ(s) t−s Mellin
B[f(t)](s) = ∫ ∞
B[f(e−t)](s) =
=
f(η) ηs−1 dη
= M[f ](s− 1)
η = e−t t : −∞ → ∞ η : ∞ → 0 dη = −e−tdt Wolfram B[f(e−t)](s) = M[f ](s) M[e−t](s)

f(t) = ∑ n
an en(t)
generating function z-
2.1
{an}n=0,1,... n tn
f(t) = ∞∑ n=0
2
n t = 0
an = f (n)(0)

{e2πint}n∈Z {an}n∈Z
f(t) = ∞∑
n=−∞
an e 2πint (4)
f(t) 1 [0, 1) {e2πint}n∈Z an
an =
∫ 1
0
f(t) e−2πint dt (5)
Fourier 1807 1 f(t) (5) (4) 1904 L2([0, 1)) 1845 “Uber die Darstellbarkeit einer Function durch eine trigonometrische Reihe”

2.3
L[f ](s) = ∫ ∞
e−st = ∞∑ n=0
3
∫ ∞
3 Mellin fα(t) = f(αt) Mellin
Mfα(ω) =
.
1 sin(t) sin(αt), α = 0.7 0.2 Mellin α ω
0 2 4 6 8 10t -1
-0.5
0
0.5
1
0.7
0.702
0.704
0.706
0.708
a
w
1: sin(t) sin(αt), α = 0.7 0.2α
t = 0 2
4
0 5 10 15 20t -1
-0.5
0
0.5
1
0.64
0.66
0.68
0.7
0.72
0.74
-0.5
0
0.5
1
0.7
0.705
0.71
0.715
0.72
0.725
1 Mellin f(t) fα(t) = f(αt)
f(ξ) =
fα(ξ) =
α =
1
5
Mellin p > 0
Mellin
α
M f (p, ω) =⇒ α =
M fα (p, ω)
M f (p, ω)
4.1.1
3 2 [0, 2π] sin(2πt) 2 1 = 0.01 α = 1.3 sin(2παt) 2 [0, 3π] 2 = 0.007 fft 0
fft Matlab fftshift
• −π/1
• 2π/1/
6
0 2 4 6 8 10 -1
-0.5
0
0.5
0.2
0.4
0.6
0.8
1
[rad]
3: α = 1.3 [−100, 100] [rad]
3 p α 4 5
p = 3 α = 1.3 ω = p− 1
-100 -50 0 50 100 0
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.18
1.2
1.22
1.24
1.26
1.28
w
a
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.05
1.1
1.15
1.2
1.25
1.3
w
a
4: p = 1, 2 α = 1.3 ω
7
-100 -50 0 50 100 0
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.05
1.1
1.15
1.2
1.25
1.3
w
a
0.2
0.4
0.6
0.8
1
[rad] 0 2 4 6 1.297
1.298
1.299
1.3
1.301
w
a
5: p = 3, 4 α = 1.3 ω
5 2 Mellin Mellin 1 [0,∞)
2 (r, θ) θ r Mellin Mellin t ω f(t)
2 Mellin
5.1 Mellin
6 (r, θ) θ 0 ≤ θ ≤ 90 7 α = 0.73 θ 0
90 0.5 ω 1 10 0.1 α 8 θ 0 90 ω ω = 1 ω = 10 α 0.71
0.74 8 α
8
r
q
r
O
6: (r, θ) ×
7: 0.73 0.73 0.91
7 α1 = 0.73 α2 = 0.91 α 9
5.2 2 Mellin
R2 f(x1, x2) Mellin
Mf(ω) =
∫ R2
2) ω/2dx1dx2
9
a = 0.73
1 10w
0
45
90
0.72
0.725
0.73
0.735
0.74
q
5
8: 0.73 Mellin α θω
α
a = 0.73, a = 0.91
1 10w5
1 2
9: 0.73 0.91 Mellin α
θω α α1 = 0.73
α2 = 0.91
Mfα(ω) =
∫ R2
2) ω/2dx1dx2
10
α
ω + 2
7 0.73 α 10 1 15 ω α = 0.73
0 5 10 15
w
10: 0.73 Mellin α ω α = 0.73
6 2 Mellin 2 f(x1, x2)
f(ξ1, ξ2) =
fα(x1, x2) = f(αx1, αx2)y1 = αx1,
y2 = αx1 dy1dy2 = α2dx1dx2
fα(ξ1, ξ2) =
=
dy1dy2 α2
11
f(ξ1, ξ2) f(ξ1, ξ2) p > 0 Mellin

f(ξ1, ξ2)p (ξ21 + ξ22) ω/2dξ1dξ2
M|fα| (p, ω) = ∫ R2
fα(ξ1, ξ2)p (ξ21 + ξ22) ω/2dξ1dξ2
=
)ω/2 α2dη1dη2
= αω+2−2pM|f | (p, ω)
η1 = ξ1/α, η2 = ξ2/α, dη1dη2 = dξ1dξ2/α 2 α
M|f | (p, ω) M|fα| (p, ω)
αω+2−2p = M|fα| (p, ω) M|f | (p, ω)
=⇒ α =
) 1
AIMaP (C) 17K05363
[1] R. Godement, Analysis III, Analytic and Differential Functions, Manifolds
and Riemann Surfaces, Springer, 2015.
[2] Wolfram MathWorld Mellin Transform http://mathworld.
wolfram.com/MellinTransform.html 2019 12 5
wiki/Mellin_transform 2019 12 5
12
wavelet2019
wavelet2019programchair2
1_toda
2_AIMap2019aki_v006ikawa
3_Arai_New
4_MorimotoTR

ウェーブレット理論と工学への応用 - AIMaP...The original chromatic scale All the...

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Transcript of ウェーブレット理論と工学への応用 - AIMaP...The original chromatic scale All the...